The 20-run victory at Shere Bangla National Stadium came five months after Bangladesh secured their first test win over Sri Lanka and a month before the first anniversary of their maiden triumph over England in the longest form of the game.
Mushfiqur said the first innings partnership of 155 between Man-of-the-Match Shakib Al Hasan and opener Tamim Iqbal had been crucial to the victory with the former also taking 5-85 with his spin-bowling to help wrap up the win on Wednesday.
"Obviously a great feeling, beating Australia. A great effort by the boys, especially Shakib and Tamim were outstanding," Mushfiqur said at the post-match presentation.
"If you look at our home performance, it has been pretty consistent. Against England (last year) also, we were very close in the first test and beat them in the second test. That's the belief we have.
"I think the spinners did a great job. I thought Taijul (Islam), Miraj (Mehidy Hasan) and Shakib bowled really well in both the innings (but) I think Tamim-Shakib partnership in the first innings set up the game for us."
Mushfiqur said the lower order woes that prevented Bangladesh from driving home their first-innings advantage showed there was plenty of room for improvement.
"The lower order is working hard and if you see at the last few series, they are much better now," he said.
"In this test, the wicket was pretty tough to bat on, that has to be said, because the top order also found it difficult to bat.
"The boys showed character. All the credit goes to the bowlers, and fielders. They did miss some (catches) but it can happen. Hopefully we'll come harder at them in the next test and hopefully we'll seal the series."
Australia did well to make the contest so close after starting the second innings needing 265 to win on a deteriorating pitch.
David Warner top-scored with a brilliant 112 but captain Steve Smith said they ultimately paid the price for a poor performance in the first innings, when they made 217.
"It was a great test match, credit to Bangladesh the way they played," he said.
"In the first innings the partnership between Tamim and Shakib really set the game up for them and then I thought we really fought back well after that.
"We probably let ourselves down in the first innings with the bat, probably should have made a few more there. It was pretty difficult batting here in the second innings and I thought we did pretty well to get 240 in the end.
"It's set it up for a really good series."
Chittagong hosts the second and final test from Monday.
(Reporting by Nick Mulvenney in Sydney,; Editing by Amlan Chakraborty)
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